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A274394
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E.g.f. A(x) satisfies: A( A( x^4*exp(-4*x) )^(1/4) ) = x.
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4
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1, 2, 9, 64, 595, 7416, 111979, 1989632, 40695561, 941667040, 24323649361, 693818707968, 21661372820971, 734712173277824, 26902827107293635, 1057724890214957056, 44442356900221356241, 1987370544970750468608, 94240073170115929379161, 4723448516579307027169280, 249510355552473169494452931, 13854414947224528743034304512, 806733172355775780726416256859
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( A( x^4*exp(4*x) )^(1/4) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( A( x^4*exp(-4*x) )^(1/4) ).
(3) A( A(x)^4 * exp(-4*A(x)) ) = x^4.
(4) A(-A(x)^4 * exp(-4*A(x)) ) = -LambertW(x^4*exp(-x^4)).
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 595*x^5/5! + 7416*x^6/6! + 111979*x^7/7! + 1989632*x^8/8! + 40695561*x^9/9! + 941667040*x^10/10! + 24323649361*x^11/11! + 693818707968*x^12/12! + 21661372820971*x^13/13! + 734712173277824*x^14/14! + 26902827107293635*x^15/15! + 1057724890214957056*x^16/16! +...
such that A( A( x^4*exp(-4*x) )^(1/4) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^4*exp(-4*x) )^(1/4) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 35*x^5/5! - 906*x^6/6! + 15757*x^7/7! - 210008*x^8/8! + 2464569*x^9/9! - 32810410*x^10/10! + 671239811*x^11/11! - 18224632812*x^12/12! + 496597765963*x^13/13! - 12681217528994*x^14/14! + 320976165059565*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^4 * exp(-4*A(x)) = x^4 - 2*x^8/2! + 3*x^12/3! - 4*x^16/4! + 35*x^20/5! - 906*x^24/6! + 15757*x^28/7! - 210008*x^32/8! + 2464569*x^36/9! - 32810410*x^40/10! +...
where A( A(x)^4 * exp(-4*A(x)) ) = x^4.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( A(x^4*exp(4*x))^(1/4) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^4*exp(-4*x +x*O(x^n)))^(1/4) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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